Let $R = \Bbbk[x_1, \dots, x_n]/(p)$, where $\Bbbk$ is an algebraically closed field of characteristic zero and where $p \in \Bbbk[x_1, \dots, x_n]$ is prime, so that $R$ is a domain. In general, $R$ is not a UFD: one such example is $\Bbbk[w,x,y,z]/(wx-yz)$, where $wx=yz$ has two factorizations into irreducibles. Obviously, one can invert every element of $R$ to get a field, and hence a UFD, but I'm interested in only inverting a subset of elements of $R$:
Let $p \in \Bbbk[x_1, \dots, x_n]$ be prime. Is $\Bbbk[x_1^{\pm 1}, \dots, x_n^{\pm 1}]/(p)$ a UFD? If this isn't true in general, can any simple hypotheses be placed on $p$ to guarantee that it is a UFD?
No, $k[x_1^{\pm 1}, \dots, x_n^{\pm 1}]/(p)$ is not a UFD in general.
For example if $p(x,y)=(y-1)^2-(x-1)^3\in k[x^{\pm 1},y^{\pm 1}]$, then the quotient ring $R=k[x^{\pm 1},y^{\pm 1}]/(p)$ is not a UFD, as witnessed by the two factorizations of $(\bar y-1)^2=(\bar x-1)^3\in R$ ( There are details to check, like irreducibility of $\bar y-1$ and $\bar x-1$).